The coefficient of

Question:

The coefficient of $x^{4}$ in the expansion of $\left(1+x+x^{2}+x^{3}\right)^{6}$ in powers of $x$, is__________.

Solution:

Coefficient of $x^{4}$ in $\left(\frac{1-x^{4}}{1-x}\right)^{6}=$ coefficient of $x^{4}$ in

$\left(1-6 x^{4}\right)(1-x)^{-6}$

$=$ coefficient of $x^{4}$ in $\left(1-6 x^{4}\right)\left[1+{ }^{6} C_{1} x+{ }^{7} C_{2} x^{2}+\ldots . .\right]$

$={ }^{9} C_{4}-6 \cdot 1=126-6=120$

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